Homomorphic images of polynomial near-rings. (Q1005905)

The term ``polynomial near-ring'' has (presently) 3 meanings: ``ordinary'' polynomials over rings (usually commutative with identity) w.r.t. addition and composition, polynomials with coefficients in near-rings in the variety of near-rings in the sense of Lausch-Nöbauer, and thirdly in the sense of A. van der Walt as special cases of group near-rings. All these concepts lead to near-rings with their own flavour. This paper deals with the third version. Starting with a zero-symmetric right near-ring \(N\) with identity, the author considers the near-ring \(M_N(N^k)\) of all maps \(f\) from \(N^k\) into itself with \(f(gn)=f(g)n\) for all \(n\in N\) and \(g\in N^k\). \(N\) can be embedded in \(M_N(N^k)\) and elements in the complement are called ``indeterminates''. If \(x\) is such an indeterminate, the subnear-ring of \(M_N(N^k)\) generated by \(N\) and \(x\) is denoted by \(N[x]\), called the polynomial near-ring over \(N\), and studied extensively. In particular, quotients of \(N[x]\) w.r.t. principal ideals are considered; they often are again polynomial near-rings. \(N[x]\) can also be viewed as a subnear-ring of an infinite matrix near-ring. More than one variables are also studied, and many examples are given. It is natural for this context that many parts of the paper are rather technical.